(sec:surfreacmodel)=
# Surface Reactions Models

Heterogeneous surface reactions are described using the hydrogen-abstraction–acetylene-addition (HACA) mechanism. Soot growth in the HACA mechanism proceeds through a sequence similar to PAH growth. Hydrogenated armchair sites, $\mathrm{C}_{\mathrm{soot}}{-}\mathrm{H}$, located at the edges of aromatic structures are dehydrogenated through hydrogen abstraction to form radical sites, $\mathrm{C}_{\mathrm{soot}}^{\circ}$. These radical sites react with $\mathrm{C_2H_2}$, producing an additional aromatic ring with hydrogenated surface sites.

The radical and hydrogenated surface sites can also react with $\mathrm{O_2}$ and $\mathrm{OH}$, respectively, resulting in the removal of carbon from soot particles through oxidation. The elementary reactions used to describe these processes are listed in {numref}`tab:HACA`.

The soot mass-growth rate through HACA is obtained from the reaction of $\mathrm{C_2H_2}$ with dehydrogenated surface sites:

```{math}
:label: eqn:hacaRate

\omega_{\mathrm{gr}}^i
=
\alpha^i
k_{f,4}
[\mathrm{C_2H_2}]
[\mathrm{C}_{\mathrm{soot}}^{\circ,i}].
```

Here, $k_{f,4}$ is the forward rate coefficient of Reaction {eq}`reac:haca4`. The concentration of dehydrogenated sites, $[\mathrm{C}_{\mathrm{soot}}^{\circ,i}]$, is obtained by multiplying the surface density of dehydrogenated sites by the total soot surface area per unit mass of gas mixture in section $i$:

```{math}
:label: eqn:csoot0

[\mathrm{C}_{\mathrm{soot}}^{\circ,i}]
=
\frac{\rho}{Av}
A_{\mathrm{tot}}^i
\chi_{\mathrm{soot}}^{\circ}.
```

The surface density of dehydrogenated sites, $\chi_{\mathrm{soot}}^{\circ}$, is calculated by applying a steady-state approximation to $[\mathrm{C}_{\mathrm{soot}}^{\circ}]$ for the reaction system listed in {numref}`tab:HACA`:

```{math}
:label: eqn:chisoot0

\chi_{\mathrm{soot}}^{\circ}
=
\frac{
k_{f,1}[\mathrm{H}]
+
k_{f,2}[\mathrm{OH}]
}{
k_{r,1}[\mathrm{H_2}]
+
k_{r,2}[\mathrm{H_2O}]
+
k_{f,3}[\mathrm{H}]
+
k_{f,4}[\mathrm{C_2H_2}]
+
k_{f,5}[\mathrm{O_2}]
}
\chi_{\mathrm{soot-H}}.
```

The surface density of hydrogenated sites, $\chi_{\mathrm{soot-H}}$, is estimated by assuming that the soot surface is composed of outward-facing PAH edges assembled into turbostratic structures {cite:p}`frenklach2019new`. Using an interlayer spacing of 3.15 $\mathrm{\mathring{A}}$ and two C–H bonds per benzene-ring length gives

```{math}

\chi_{\mathrm{soot-H}}
=
0.23\ \mathrm{site\,\mathring{A}^{-2}}
=
2.3\times10^{19}\ \mathrm{site\,m^{-2}},
```

which represents the maximum theoretical surface-site density.

In Equation {eq}`eqn:hacaRate`, $\alpha^i$ is the surface-reactivity factor. It ranges from 0 to 1 and represents the reduction in the number of available reaction sites relative to the theoretical maximum because of PAH-layer orientation, particle aging, surface growth, and soot maturity {cite:p}`haynes1982surface,harris1985chemical`. The surface-reactivity factor has also been observed to depend on the temperature–time history of soot particles {cite:p}`homann1985formation,dasch1985decay`.

The value of $\alpha$ has been represented using constant, application-specific values and empirical expressions based on particle size and flame temperature. A detailed review is provided in Chapter 4 of {cite:t}`veshkini2015understanding`. Omnisoot can calculate $\alpha^i$ using the empirical expression proposed by {cite:t}`appel2000kinetic`:

```{math}
:label: eqn:alpha

\alpha^i
=
\tanh
\left[
\frac{
12.56-0.00563T
}{
\log_{10}
\left(
\frac{\rho_{\mathrm{soot}}Av}{W_{\mathrm{carbon}}}
\frac{\pi}{6}
\left(d_p^i\right)^3
\right)
}
-
1.38
+
0.00068T
\right].
```

Alternatively, $\alpha^i$ can be related to the H/C ratio of soot particles by assuming that all hydrogen atoms reside on the particle surface {cite:p}`blanquart2009joint`:

```{math}
:label: eqn:alpha_htoc

\alpha^i
=
\frac{
H_{\mathrm{tot}}^i
}{
C_{\mathrm{tot}}^i
}.
```

The HACA contributions to the carbon and hydrogen source terms are calculated from the HACA growth rate by accounting for the two carbon atoms in $\mathrm{C_2H_2}$ and the relative numbers of armchair and zigzag hydrogenated sites on the soot surface {cite:p}`blanquart2009analyzing`:

```{math}
:label: eqn:IiCtotgr

I_{C_{\mathrm{tot}},\mathrm{haca}}^i
=
\frac{
2\omega_{\mathrm{gr}}^i
}{
\rho
}.
```

```{math}
:label: eqn:IiHtotgr

I_{H_{\mathrm{tot}},\mathrm{haca}}^i
=
\frac{
0.25\omega_{\mathrm{gr}}^i
}{
\rho
}.
```

The rates of change of the concentrations of $\mathrm{C_2H_2}$ and H radicals due to HACA growth are

```{math}
:label: eqn:C2H2rate_gr

\left(
\frac{
\mathrm{d}[\mathrm{C_2H_2}]
}{
\mathrm{d}t
}
\right)_{\mathrm{gr}}
=
-\sum_{i=1}^{n_{\mathrm{sec}}}
\omega_{\mathrm{gr}}^i.
```

```{math}
:label: eqn:Hrate_gr

\left(
\frac{
\mathrm{d}[\mathrm{H}]
}{
\mathrm{d}t
}
\right)_{\mathrm{gr}}
=
1.75
\sum_{i=1}^{n_{\mathrm{sec}}}
\omega_{\mathrm{gr}}^i.
```

The HACA surface reactions are

```{math}
:label: reac:haca1

\mathrm{C}_{\mathrm{soot}}{-}\mathrm{H}
+
\mathrm{H}
\overset{k_{f,1}}{
\underset{k_{r,1}}{\rightleftharpoons}
}
\mathrm{C}_{\mathrm{soot}}^{\circ}
+
\mathrm{H_2}.
```

```{math}
:label: reac:haca2

\mathrm{C}_{\mathrm{soot}}{-}\mathrm{H}
+
\mathrm{OH}
\overset{k_{f,2}}{
\underset{k_{r,2}}{\rightleftharpoons}
}
\mathrm{C}_{\mathrm{soot}}^{\circ}
+
\mathrm{H_2O}.
```

```{math}
:label: reac:haca3

\mathrm{C}_{\mathrm{soot}}^{\circ}
+
\mathrm{H}
\overset{k_{f,3}}{\longrightarrow}
\mathrm{C}_{\mathrm{soot}}{-}\mathrm{H}.
```

```{math}
:label: reac:haca4

\mathrm{C}_{\mathrm{soot}}^{\circ}
+
\mathrm{C_2H_2}
\overset{k_{f,4}}{\longrightarrow}
\mathrm{C}_{\mathrm{soot}}{-}\mathrm{H}
+
\mathrm{H}.
```

```{math}
:label: reac:haca5

\mathrm{C}_{\mathrm{soot}}^{\circ}
+
\mathrm{O_2}
\overset{k_{f,5}}{\longrightarrow}
2\mathrm{CO}
+
\mathrm{product}.
```

```{math}
:label: reac:haca6

\mathrm{C}_{\mathrm{soot}}{-}\mathrm{H}
+
\mathrm{OH}
\overset{k_{f,6}}{\longrightarrow}
\mathrm{CO}
+
\mathrm{product}.
```

```{list-table} Arrhenius rate coefficients for the HACA surface reactions, $k=AT^n\exp[-E/(RT)]$.
:header-rows: 1
:name: tab:HACA
:widths: 12 40 12 20 8 14

* - Reaction
  - Pathway
  - Direction
  - $A$ [$\mathrm{m^3\,mol^{-1}\,s^{-1}}$]
  - $n$
  - $E/R$ [K]
* - {eq}`reac:haca1`
  - $\mathrm{C}_{\mathrm{soot}}{-}\mathrm{H}+\mathrm{H}\rightleftharpoons\mathrm{C}_{\mathrm{soot}}^{\circ}+\mathrm{H_2}$
  - Forward
  - $4.17\times10^7$
  - 0
  - 6542.52
* - {eq}`reac:haca1`
  - 
  - Reverse
  - $3.9\times10^6$
  - 0
  - 5535.98
* - {eq}`reac:haca2`
  - $\mathrm{C}_{\mathrm{soot}}{-}\mathrm{H}+\mathrm{OH}\rightleftharpoons\mathrm{C}_{\mathrm{soot}}^{\circ}+\mathrm{H_2O}$
  - Forward
  - $1.0\times10^4$
  - 0.734
  - 719.68
* - {eq}`reac:haca2`
  -
  - Reverse
  - $3.68\times10^2$
  - 1.139
  - 8605.94
* - {eq}`reac:haca3`
  - $\mathrm{C}_{\mathrm{soot}}^{\circ}+\mathrm{H}\rightarrow\mathrm{C}_{\mathrm{soot}}{-}\mathrm{H}$
  - Forward
  - $1.0\times10^4$
  - 0.734
  - 719.68
* - {eq}`reac:haca4`
  - $\mathrm{C}_{\mathrm{soot}}^{\circ}+\mathrm{C_2H_2}\rightarrow\mathrm{C}_{\mathrm{soot}}{-}\mathrm{H}+\mathrm{H}$
  - Forward
  - 80
  - 1.56
  - 1912.43
* - {eq}`reac:haca5`
  - $\mathrm{C}_{\mathrm{soot}}^{\circ}+\mathrm{O_2}\rightarrow2\mathrm{CO}+\mathrm{product}$
  - Forward
  - $2.2\times10^6$
  - 0
  - 3774.53
* - {eq}`reac:haca6`
  - $\mathrm{C}_{\mathrm{soot}}{-}\mathrm{H}+\mathrm{OH}\rightarrow\mathrm{CO}+\mathrm{product}$
  - Forward
  - $\gamma_{\mathrm{OH}}=0.13$
  - —
  - —
```

Carbon atoms on the soot surface are oxidized through reactions with $\mathrm{O_2}$ and $\mathrm{OH}$, represented by Reactions {eq}`reac:haca5` and {eq}`reac:haca6`, respectively. These pathways decrease the total carbon content of soot and release gaseous products.

The $\mathrm{O_2}$- and $\mathrm{OH}$-oxidation rates are

```{math}
:label: eqn:hacaO2Rate

\omega_{\mathrm{ox,O_2}}^i
=
\alpha^i
k_{f,5}
[\mathrm{O_2}]
[\mathrm{C}_{\mathrm{soot}}^{\circ,i}].
```

```{math}
:label: eqn:hacaOHRate

\omega_{\mathrm{ox,OH}}^i
=
\gamma_{\mathrm{OH}}
\beta_{\mathrm{OH}}^i
Av
[\mathrm{OH}]
[\mathrm{soot}^i].
```

Here, $\gamma_{\mathrm{OH}}=0.13$ is the reaction probability for collisions between OH radicals and soot particles {cite:p}`appel2000kinetic`. The collision frequency between OH and soot particles, $\beta_{\mathrm{OH}}^i$, is calculated from kinetic theory:

```{math}
:label: eqn:betaOH

\beta_{\mathrm{OH}}^i
=
\sqrt{
\frac{\pi k_B T}{2}
\left(
\frac{1}{m_{\mathrm{agg}}^i}
+
\frac{1}{m_{\mathrm{OH}}}
\right)
}
\left(
d_c^i+d_{\mathrm{OH}}
\right)^2.
```

The mass and equivalent diameter of an OH radical are $m_{\mathrm{OH}}=2.824\times10^{-26}$ kg and $d_{\mathrm{OH}}=0.3$ nm, respectively {cite:p}`shepherd2022measurement`.

The oxidation contribution to the total-carbon source term is calculated by accounting for the number of carbon atoms removed through each pathway:

```{math}
:label: eqn:ICtot

I_{C_{\mathrm{tot}},\mathrm{ox}}^i
=
\frac{
2\omega_{\mathrm{ox,O_2}}^i
+
\omega_{\mathrm{ox,OH}}^i
}{
\rho
}.
```

The rates of change of the concentrations of $\mathrm{CO}$, $\mathrm{O_2}$, $\mathrm{OH}$, and H due to oxidation are

```{math}
:label: eqn:COrate_ox

\left(
\frac{
\mathrm{d}[\mathrm{CO}]
}{
\mathrm{d}t
}
\right)_{\mathrm{ox}}
=
2
\sum_{i=1}^{n_{\mathrm{sec}}}
\omega_{\mathrm{ox,O_2}}^i.
```

```{math}
:label: eqn:O2rate_ox

\left(
\frac{
\mathrm{d}[\mathrm{O_2}]
}{
\mathrm{d}t
}
\right)_{\mathrm{ox}}
=
-\sum_{i=1}^{n_{\mathrm{sec}}}
\omega_{\mathrm{ox,O_2}}^i.
```

```{math}
:label: eqn:Hrate_ox

\left(
\frac{
\mathrm{d}[\mathrm{OH}]
}{
\mathrm{d}t
}
\right)_{\mathrm{ox}}
=
-\sum_{i=1}^{n_{\mathrm{sec}}}
\omega_{\mathrm{ox,OH}}^i.
```

```{math}
:label: eqn:OHrate_ox

\left(
\frac{
\mathrm{d}[\mathrm{H}]
}{
\mathrm{d}t
}
\right)_{\mathrm{ox}}
=
\sum_{i=1}^{n_{\mathrm{sec}}}
\omega_{\mathrm{ox,OH}}^i.
```